Inverse Source and Inverse Scattering Problems

逆源和逆散射问题

• 批准号: 1309362
• 负责人: John Sylvester
• 金额: $ 20.91万
• 依托单位: University of Washington
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-08-01 至 2017-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1309362&HistoricalAwards=false
• 关键词: Inverse; Source; Scattering; Problems

The essential feature of remote sensing is that all information must be transported through an intervening medium by (electromagnetic or acoustic) waves. We observe only the scattered field, the wave after it has passed through the intervening medium (which may be empty space). The simplest mathematical model is the scalar wave equation, which decomposes, via Fourier transform, into the scalar Helmholtz equation at each wavenumber. The phenomenon of evanescence limits resolution by allowing only a finite dimensional family of waves to reach the sensors with an amplitude that has not been severely attenuated. The dimension of that family gives important information about the size of the source of the wave, and the number of nonzero coefficients needed to represent the wave in different bases (Hankel function expansions centered at different points) carries information about the location of the source or sources. We are developing methods to use this information. This is complicated by the fact that the source is not uniquely determined by the scattered wave. We resolve this by seeking the smallest set that supports (carries) a source that radiates that scattered field, and must have been part of any other source that radiated the same field. Examples show that there is no such set in general, but there is a smallest union of well separated convex sets that satisfies both criteria. A major goal of this project is to find an algorithm to find this union of well separated convex sets, and give reliable estimates on how far apart sources of various sizes must be in order to guarantee that the individual sources will be visible in the data. This analysis is relevant to many other inverse scattering problems. A particularly important application is understanding the circumstances in which the Born, or single scattering, approximation is a reliable substitute for the full scattering problem. While all practitioners of inverse scattering techniques know that the farther apart two scatterers are, the less they interact, and therefore the better the Born approximation works, current mathematical theory seems to say the opposite. The apparent paradox is due to the norms we use to measure and compare the waves. Most mathematically tractable norms have features which misrepresent some of the physics. We continue to work to formulate norms which are simple enough to allow mathematical analysis and yet faithfully reflect physical reality.A remote sensing experiment gathers information about an object from a distance, without any direct contact. An underwater array of acoustic sensors that seeks to locate a submarine (the source) based on the noise radiated from the submarine's engine, is an example of passive remote sensing system. While a sonar array that transmits its own wave in order to locate the submarine based on the properties of the echo, is an example of active remote sensing system. In this case the array, and not the submarine, is the primary source, and the submarine is referred to as the scatterer.The inverse source and scattering problems are an essential part of remote sensing, and algorithms for finding small point-like sources and scatterers have had major impact on technological development, including antenna design. The project aims to bridge the gap between effective algorithms and theory by reformulating the analysis in a way that more accurately reflects the way this algorithms work with finite data sets.

遥感的基本特征是，所有信息都必须通过（电磁波或声波）中介媒介传输。我们只观察到散射场，即波穿过中间介质（可能是空的空间）后的波。最简单的数学模型是标量波动方程，它通过傅里叶变换分解为每个波数的标量亥姆霍兹方程。 倏逝现象通过仅允许有限维波族以尚未严重衰减的振幅到达传感器来限制分辨率。该族的维数给出了关于波源大小的重要信息，并且在不同基（以不同点为中心的汉克尔函数展开）中表示波所需的非零系数的数量携带了关于源或源的位置的信息。我们正在开发使用这些信息的方法。这是复杂的事实，即源不是唯一确定的散射波。我们通过寻找支持（携带）辐射该散射场的源的最小集合来解决这个问题，并且必须是辐射相同场的任何其他源的一部分。例子表明，一般不存在这样的集合，但存在一个满足这两个标准的良好分离的凸集的最小并集。这个项目的一个主要目标是找到一个算法来找到这个良好分离的凸集的并集，并给出可靠的估计，以保证各个源在数据中可见。这种分析与许多其他逆散射问题有关。一个特别重要的应用是了解玻恩或单次散射近似是完全散射问题的可靠替代品的情况。尽管所有逆散射技术的从业者都知道，两个散射体之间的距离越远，它们之间的相互作用就越小，因此玻恩近似的效果就越好，但目前的数学理论似乎是相反的。这个明显的悖论是由于我们用来测量和比较波浪的标准。大多数数学上易于处理的规范都有一些特征，这些特征歪曲了一些物理学。我们继续努力制定规范，这些规范简单到可以进行数学分析，但又忠实地反映了物理现实。遥感实验从远处收集关于物体的信息，而没有任何直接接触。水下声传感器阵列是被动遥感系统的一个例子，它根据潜艇发动机辐射的噪声来定位潜艇（声源）。而声纳基阵发射自己的波，以便根据回波的特性来定位潜艇，是主动遥感系统的一个例子。在这种情况下，阵列，而不是潜艇，是主要的来源，和潜艇被称为散射体。反源和散射问题是遥感的一个重要组成部分，和算法寻找小点状源和散射体有重大影响的技术发展，包括天线设计。该项目的目的是弥合有效的算法和理论之间的差距，通过重新制定的分析方式，更准确地反映了这种算法与有限的数据集的工作方式。